Mirror Quarks

The conversation regarding the new CDF paper reaches the inevitable consensus that the measured mass difference between top and antitop quarks should be ignored, just like most other inconvenient experimental results, in the purity of the dictum: stringers can doeth no wrong. Meanwhile, for the record, let us observe that our postdiction of a $0.1$ MeV mass difference between neutrons and antineutrons is at least confirmed by the rough scaling law

$\frac{3 \Delta_t}{3 \Delta_u} = 10^{5} = \frac{10^4}{0.1}$

where $\Delta_t$ and $\Delta_u$ are the mass differences for top and up quarks respectively. Henceforth, it would seem advisable to rename antiquarks as mirror quarks. (I am already having nightmares about notational dilemmas.) This most fascinating possibility immediately brings to mind the greatest paradox of the Standard Model: the neutral pion. Touted as a superposition of $u \overline{u}$ and $d \overline{d}$ pairs, the short lived $\pi^{0}$ is nonetheless not just a pair of photons. It has a mass of about $135$ MeV.

In the Standard Model, neutral pion decay results from a chiral symmetry breaking associated to a fermion triangle diagram. This kind of symmetry breaking is the result of a classical symmetry being destroyed upon quantization. In 1967, Veltman published the first important paper, examining the possibility that partial conservation of the relevant current was possible. With this hypothesis, the vanishing of the $\pi^{0}$ mass is equated with the vanishing of the $\pi^{0} \rightarrow \gamma \gamma$ decay. This is the limit of exact chiral symmetry. However, Bell and Jackiw soon noted that this disagreed with the perturbation theory of the time, and so a natural modification of the perturbation theory was constructed to properly take the anomaly into account. Neutral pion decay is then a direct result of the regularization scheme, and chiral symmetry is only a property of the classical Lagrangian.

Many have since considered this situation to be an indication of CPT or Lorentz violation. However, the modern topological underpinnings of such anomalies suggest instead a deeper, less obvious resolution to their mysterious meaning. In particular, the index calculation of the anomaly is intriguingly analogous to the derivation of $3$ generations from the index associated to the six dimensional moduli space of the $6$ punctured sphere.

Under the non local mirror quark hypothesis it is not so surprising that the neutral pion exists, because $\overline{u}$ is no longer the antiparticle of the $u$ quark. The hypothesis that $\pi^{0}$ is its own antiparticle, with preservation of Lorentz symmetry, would provide a relation between mirror quarks and antiparticles. However, it is perhaps more natural to consider the now established hypothesis that the charged pion $\pi^{+}$ is the antiparticle of $\pi^{-}$, in which case the required charge $1$ condition is

$u^{m} d = \overline{u d^{m}}$

which we might rewrite as $u u^{m}d d^{m} = \gamma$.